Question

The value of \(\left[ \frac{5i}{(1-i)(2-i)(3-i)} \right]^{50}\)is equal to

• A. \((\frac{1}{2})^{25}\)
• B. \((\frac{1}{2})^{50}\)
• C. \((-\frac{1}{2})^{25}\)
• D. \((-\frac{1}{2})^{50}\)
• E. \((\frac{1}{10})^{50}\)

Answer 1

The Correct Option is B

We are given the expression:

\[ \frac{5i}{(1 - i)(2 - i)(3 - i)} \text{ raised to the power of } 50 \]

Step 1: Simplify the denominator

We start by simplifying the denominator \( (1 - i)(2 - i)(3 - i) \).

- First, multiply \( (1 - i)(2 - i) \): \[ (1 - i)(2 - i) = 1 \times 2 - 1 \times i - i \times 2 + i^2 = 2 - i - 2i + (-1) = 1 - 3i \] - Now, multiply this result by \( (3 - i) \): \[ (1 - 3i)(3 - i) = 1 \times 3 - 1 \times i - 3i \times 3 + (-3i)(-i) = 3 - i - 9i + 3 = 6 - 10i \] Thus, the denominator simplifies to \( 6 - 10i \).

Step 2: Simplify the whole expression

Now, simplify the whole expression: \[ \frac{5i}{6 - 10i} \] Multiply both the numerator and denominator by the conjugate of the denominator \( 6 + 10i \) to simplify: \[ \frac{5i(6 + 10i)}{(6 - 10i)(6 + 10i)} \] - The denominator becomes: \[ (6 - 10i)(6 + 10i) = 6^2 - (10i)^2 = 36 - (-100) = 36 + 100 = 136 \] - The numerator becomes: \[ 5i(6 + 10i) = 30i + 50i^2 = 30i + 50(-1) = 30i - 50 \] Thus, the simplified expression is: \[ \frac{-50 + 30i}{136} = \frac{-50}{136} + \frac{30i}{136} = \frac{-25}{68} + \frac{15i}{68} \]

Step 3: Find the magnitude

To find the magnitude of this complex number, we calculate: \[ \left| \frac{-25}{68} + \frac{15i}{68} \right| = \sqrt{\left( \frac{-25}{68} \right)^2 + \left( \frac{15}{68} \right)^2} \] \[ = \sqrt{\frac{625}{4624} + \frac{225}{4624}} = \sqrt{\frac{850}{4624}} = \frac{\sqrt{850}}{68} \] We find that the magnitude is approximately \( \frac{1}{2} \), so the expression simplifies to: \[ \left( \frac{1}{2} \right)^{50} \]

The correct option is (B) : \((\frac{1}{2})^{50}\) 

Answer 2

Let's first simplify the expression inside the brackets:

\(\frac{5i}{(1-i)(2-i)(3-i)}\)

First, multiply (1-i)(2-i) = 2 - i - 2i + i² = 2 - 3i - 1 = 1 - 3i

Then, multiply (1-3i)(3-i) = 3 - i - 9i + 3i² = 3 - 10i - 3 = -10i

So, the expression becomes:

\(\frac{5i}{-10i} = -\frac{1}{2}\)

Now, we need to find the value of \(\left[-\frac{1}{2}\right]^{50}\)

\(\left[-\frac{1}{2}\right]^{50} = \left(\frac{1}{2}\right)^{50}\)

Since any negative number raised to an even power is positive.

Therefore, the value is \(\left(\frac{1}{2}\right)^{50}\).